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Entropy facilitated active transport
Abstract
We show how active transport of ions can be interpreted as an entropy facilitated process. In this interpretation, the pore geometry through which substrates are transported can give rise to a driving force. This gives a direct link between the geometry and the changes in Gibbs energy required. Quantifying the size of this effect for several proteins we find that the entropic contribution from the pore geometry is significant and we discuss how the effect can be used to interpret variations in the affinity at the binding site.
... As suggested in the literature, [3][4][5][6][7] confinement can influence significantly the transport properties of diffusing entities, which is also a central idea of the current research. Spatial constriction of the pore causes decrease in volume of configuration space available for molecules in solution filling the channel's interior (also for K + ions), and in consequence-a drop in entropy. ...
... This simplification of the 3D channel system to 1D stems from the great difference of time scales of the ion motion within pore's cross section (where rapid local equilibration occurs) and along the channel axis (down the electrochemical gradient). 6 On the base of changes in channel's cross section, an entropic potential ∆S(x) can be defined in the direction of K + motion (along the channel axis-x) (Fig. 3), according to the formula, 3 ...
We analyze the entropic effects of inner pore geometry changes of Kv 1.2 channel during membrane depolarization and their implications for the rate of transmembrane transport of potassium ions. We base this on the idea that spatial confinements within the channel pore give rise to entropic barriers which can both effectively affect the stability of open macroconformation and influence channel’s ability to conduct the potassium ions through the membrane. First, we calculate the differences in entropy between voltage-activated and resting states of the channel. As a template, we take a set of structures of channel pore in an open state at different membrane potentials generated in our previous research. The obtained results indicate that tendency to occupy open states at membrane depolarization is entropy facilitated. Second, we describe the differences in rates of K⁺ transport through the channel pore at different voltages based on the results of appropriate random walk simulations in entropic and electric potentials. The simulated single channel currents (I) suggest that the geometry changes during membrane depolarization are an important factor contributing to the observed flow of potassium ions through the channel. Nevertheless, the charge distribution within the channel pore (especially at the extracellular entrance) seems most prominent for the observed I/Imax relation at a qualitative level at analyzed voltages.
... [1][2][3][4][5][6][7][8][9] Particle invasion of a porous medium and drug delivery to cancerous tissues are examples where knowing the optimal conditions for transport is crucial. [10][11][12][13][14][15][16] Transport in micro-channels is virtually one-dimensional. Their irregular shapes, which in many cases can change over time, result in restrictions on the space that particles can occupy, leading to entropy variations along the length of the channels and over time. ...
It is shown that the action of an oscillating force on particles moving through a deformable-walled channel causes them to travel greater distances than in the case of a rigid channel. This increase in the transport efficiency is due to an intensification of the stochastic resonance effect observed in corrugated rigid channels, for which the response to the force is maximal for an optimal value of the thermal noise. The distances traveled by the particles are even larger when the oscillation of the micro-channel is synchronized with that of an applied transverse force and also when a constant external force is considered. The phenomenon found could be observed in the transport of particles through elastic porous media, in drug delivery to cancerous tissues, and in the passage of substrates through transporters in biological membranes. Our results indicate that an appropriate channel design and a suitable choice of applied forces lead to optimal scenarios for particle transport.
... [1][2][3][4][5][6][7][8][9] Particle invasion of a porous medium and drug delivery to cancerous tissues are examples where knowing the optimal conditions for transport is crucial. [10][11][12][13][14][15][16] Transport in micro-channels is virtually one-dimensional. Their irregular shapes, which in many cases can change over time, result in restrictions on the space that particles can occupy, leading to entropy variations along the length of the channels and over time. ...
It is shown that the action of an oscillating force on particles moving through a deformable-walled channel causes them to travel greater distances than in the case of a rigid channel. This increase in the transport efficiency is due to an intensification of the stochastic resonance effect observed in corrugated rigid channels, for which the response to the force is maximal for an optimal value of the thermal noise. The distances traveled by the particles are even larger when the oscillation of the micro-channel is synchronized with that of an applied transverse force and also when a constant external force is considered. The phenomenon found could be observed in the transport of particles through elastic porous media, in drug delivery to cancerous tissues, and in the passage of substrates through transporters in biological membranes. Our results indicate that an appropriate channel design and a suitable choice of applied forces lead to optimal scenarios for particle transport.
... Kobayashi et al. (5) discuss the feature that the entropy profile of the energy conversion in the pump is not taken into account. When we want to describe the pump function, this may become essential (22). The thermodynamic driving force for the conversion of a scalar to a vectorial process must be sought in the derivative −dμ/dγ. ...
... with the number f of degrees of freedom and the distance z from the reaction center in the funnel. Therefore, the free-energy potential has an entropic contribution going as GðzÞ ' Àk B T ln VðzÞ and the force exerted on the motor is FðzÞ ¼ À@ z GðzÞ [86][87][88]. During the release, the motor is thus propelled according to γ t dz=dt ¼ FðzÞ, giving the displacement z ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi 2fD t Δt p over the time interval Δt, where D t ¼ k B T=γ t is the diffusion coefficient corresponding to Stokes' friction coefficient γ t . ...
Colloidal motors with micrometer dimensions and no moving parts can be propelled by self-diffusiophoresis. Coupling between molecular concentration gradients generated by asymmetric surface chemical reactions and the velocity slip between colloidal particle and the surrounding fluid solution is responsible for propulsion. The interfacial properties involved in this propulsion mechanism can be described by nonequilibrium thermodynamics and statistical mechanics, disclosing the fundamental role of microreversibility in the coupling between motion and reaction. Among other phenomena, the approach predicts that propulsion by fuel consumption has the reciprocal effect of fuel synthesis by mechanical action.
... Furthermore, entropy provides a measure of the amount of coded information contained in the canonical genetic code [6], as well as a thermodynamic characterization of supercooled liquids [7,8] and amorphous materials [9,10]. Entropic effects are also crucial for mixture separation during adsorption processes [11,12], for the design of protein geometries to facilitate the active transport of ions [13], and in protein-protein binding for signal transduction and molecular recognition [14,15]. ...
We combine machine learning (ML) with Monte Carlo (MC) simulations to study the crystal nucleation process. Specifically, we use ML to infer the canonical partition function of the system over the range of densities and temperatures spanned during crystallization. We achieve this on the example of the Lennard-Jones system by training an artificial neural network using, as a reference dataset, equations of state for the Helmholtz free energy for the liquid and solid phases. The accuracy of the ML predictions is tested over a wide range of thermodynamic conditions, and results are shown to provide an accurate estimate for the canonical partition function, when compared to the results from flat-histogram simulations. Then, the ML predictions are used to calculate the entropy of the system during MC simulations in the isothermal-isobaric ensemble. This approach is shown to yield results in very good agreement with the experimental data for both the liquid and solid phases of argon. Finally, taking entropy as a reaction coordinate and using the umbrella sampling technique, we are able to determine the Gibbs free energy profile for the crystal nucleation process. In particular, we obtain a free energy barrier in very good agreement with the results from previous simulation studies. The approach developed here can be readily extended to molecular systems and complex fluids, and is especially promising for the study of entropy-driven processes.
... To answer this question we use the concept studied in detail in[4][5][6][7][8]that the spatial confinement causes an entropic barrier. On the base of changes in channel's cross section an entropic potential ΔS(x) can be defined in the direction of K+ motion (along the channel axis) (Fig. 2), according to the formula: ...
We analyze entropic effects of inner pore geometry changes of Kv 1.2 channel during membrane depolarization and their implications for the rate of transmembrane channel transport of potassium ions.
Sarcoplasmic reticulum (SR) Ca ²⁺ -ATPase transports two Ca ²⁺ ions from the cytoplasm to the SR lumen against a large concentration gradient. X-ray crystallography has revealed the atomic structures of the protein before and after the dissociation of Ca ²⁺ , while biochemical studies have suggested the existence of intermediate states in the transition between E1P⋅ADP⋅2Ca ²⁺ and E2P. Here, we explore the pathway and free energy profile of the transition using atomistic molecular dynamics simulations with the mean-force string method and umbrella sampling. The simulations suggest that a series of structural changes accompany the ordered dissociation of ADP, the A-domain rotation, and the rearrangement of the transmembrane (TM) helices. The luminal gate then opens to release Ca ²⁺ ions toward the SR lumen. Intermediate structures on the pathway are stabilized by transient sidechain interactions between the A- and P-domains. Lipid molecules between TM helices play a key role in the stabilization. Free energy profiles of the transition assuming different protonation states suggest rapid exchanges between Ca ²⁺ ions and protons when the Ca ²⁺ ions are released toward the SR lumen.
To explain the increased transport of nutrients and metabolites and to control the movement of drug molecules through the transporters to the cancer cells, it is important to understand the exact mechanism of their structure and activity, as well as their biological and physical characteristics. We propose a computational model that reproduces the functionality of membrane transporters by quantifying the flow of substrates through the cell membrane. The model identifies the force induced by conformational changes of the transporter due to hydrolysis of ATP, in ABC transporters, or by an electrochemical gradient of ions, in secondary transporters. The transport rate is computed by averaging the velocity generated by the force along the paths followed by the substrates. The results obtained are in accordance with the experiments. The model provides an overall framework for analyzing the membrane transport proteins that regulate the flows of ions, nutrients and other molecules across the cell membranes, and their activities.
Ion channels exhibit a remarkably high accuracy in selecting uniquely its associated type of ion. The mechanisms behind ion selectivity are not well understood. Current explanations build mainly on molecular biology and bioinformatics. Here we propose a simple physical model for ion selectivity based on the driven damped harmonic oscillator (DDHO). The driving force for this oscillator is provided by self-organizing harmonic turbulent structures in the dehydrating ionic flow through the ion channel, namely, oscillating pressure waves in one dimension, and toroidal vortices in two and three dimensions. Density fluctuations caused by these turbulences efficiently transmit their energy to aqua ions that resonate with the driving frequency. Consequently, these release their hydration shell and exit the ion channel as free ions. Existing modeling frameworks do not express the required complex spatiotemporal dynamics, hence we introduce a macroscopic continuum model for ionic dehydration and transport, based on the hydrodynamics of a dissipative ionic flow through an ion channel, subject to electrostatic and amphiphilic interactions. This model combines three classical physical fields: Navier-Stokes equations from hydrodynamics, Gauss's law from Maxwell theory, and the convection-diffusion equation from continuum physics. Numerical experiments with mixtures of chemical species of ions in various degrees of hydration indeed reveal the emergence of strong oscillations in the ionic flow that are instrumental in the efficient dehydration and cause a strong ionic jet into the cell. As such, they provide a powerful engine for the DDHO mechanism. Theoretical predictions of the modeling framework match significantly with empirical patch-clamp data. The DDHO standard response curve defines a unique resonance frequency that depends on the mass and charge of the ion. In this way, the driving oscillations act as a selection mechanism that filters out one specific ion. Application of the DDHO model to real ion data shows that this mechanism indeed clearly distinguishes between chemical species and between aqua and bare ions with a large Mahalanobis distance and high oscillator quality. The DDHO framework helps to understand how SNP mutations can cause severe genetic pathologies as they destroy the geometry of the channel and so alter the resonance peaks of the required ion type.
A set of disordered interacting building blocks may form ordered structures by means of a self-assembling process. An external intervention to the system by adding a chemical species or by applying forces, leads to different self-assembly scenarios with the appearance of new structures. For instance, the formation of microtubules, gels, virus capsides, cells and living beings among others takes place by self-assembly under nonequilibrium conditions. A general evolution criterion able to account for why nature selects some structures outside equilibrium and not others is lacking. Nevertheless, progress in the understanding of nonequilibrium self-assembly (NESA) mechanisms has been made thanks to the formulation of models that take particular situations into consideration. We review recent efforts devoted to describing self-assembling in out of equilibrium and we give a reference to link several current concepts in order to help in the development of new models and experimental studies. We hope that the knowledge of the intimate mechanisms leading to the formation of structures will make the implementation of re-configurable and bio-inspired materials possible as well as to give a simpler perspective to the understanding of emergence of life.
Copper and zinc are micronutrients essential for the function of many enzymes while also toxic at elevated concentrations. Cu(I)- and Zn(II)-transporting P-type ATPases of subclass 1B are of key importance for homeostasis of these transition metals, allowing ion transport across cellular membranes at the expense of ATP. Recent biochemical studies and crystal structures have significantly increased the understanding of the transport mechanisms of these proteins, but many details about their structure and function remains elusive. Here we compare the Cu(I)- and Zn(II)-ATPases, scrutinizing the molecular differences that allow transport of these two distinct metal types, and discuss possible future directions of research in the field.
Zinc is an essential micronutrient for all living organisms. It is required for signalling and proper functioning of a range of proteins involved in, for example, DNA binding and enzymatic catalysis. In prokaryotes and photosynthetic eukaryotes, Zn(2+)-transporting P-type ATPases of class IB (ZntA) are crucial for cellular redistribution and detoxification of Zn(2+) and related elements. Here we present crystal structures representing the phosphoenzyme ground state (E2P) and a dephosphorylation intermediate (E2·Pi) of ZntA from Shigella sonnei, determined at 3.2 Å and 2.7 Å resolution, respectively. The structures reveal a similar fold to Cu(+)-ATPases, with an amphipathic helix at the membrane interface. A conserved electronegative funnel connects this region to the intramembranous high-affinity ion-binding site and may promote specific uptake of cellular Zn(2+) ions by the transporter. The E2P structure displays a wide extracellular release pathway reaching the invariant residues at the high-affinity site, including C392, C394 and D714. The pathway closes in the E2·Pi state, in which D714 interacts with the conserved residue K693, which possibly stimulates Zn(2+) release as a built-in counter ion, as has been proposed for H(+)-ATPases. Indeed, transport studies in liposomes provide experimental support for ZntA activity without counter transport. These findings suggest a mechanistic link between PIB-type Zn(2+)-ATPases and PIII-type H(+)-ATPases and at the same time show structural features of the extracellular release pathway that resemble PII-type ATPases such as the sarcoplasmic/endoplasmic reticulum Ca(2+)-ATPase (SERCA) and Na(+), K(+)-ATPase. These findings considerably increase our understanding of zinc transport in cells and represent new possibilities for biotechnology and biomedicine.
We present a particle separation mechanism which induces the motion of particles of different sizes in opposite directions. The mechanism is based on the combined action of a driving force and an entropic rectification of the Brownian fluctuations caused by the asymmetric form of the channel along which particles proceed. The entropic splitting effect shown could be controlled upon variation of the geometrical parameters of the channel and could be implemented in narrow channels and microfluidic devices.
The calcium transport ATPase and the copper transport ATPase are members of the P-ATPase family and retain an analogous catalytic mechanism for ATP utilization, including intermediate phosphoryl transfer to a conserved aspartyl residue, vectorial displacement of bound cation, and final hydrolytic cleavage of Pi. Both ATPases undergo protein conformational changes concomitant with catalytic events. Yet, the two ATPases are prototypes of different features with regard to transduction and signaling mechanisms. The calcium ATPase resides stably on membranes delimiting cellular compartments, acquires free Ca(2+) with high affinity on one side of the membrane, and releases the bound Ca(2+) on the other side of the membrane to yield a high free Ca(2+) gradient. These features are a basic requirement for cellular Ca(2+) signaling mechanisms. On the other hand, the copper ATPase acquires copper through exchange with donor proteins, and undergoes intracellular trafficking to deliver copper to acceptor proteins. In addition to the cation transport site and the conserved aspartate undergoing catalytic phosphorylation, the copper ATPase has copper binding regulatory sites on a unique N-terminal protein extension, and has also serine residues undergoing kinase assisted phosphorylation. These additional features are involved in the mechanism of copper ATPase intracellular trafficking which is required to deliver copper to plasma membranes for extrusion, and to the trans-Golgi network for incorporation into metalloproteins. Isoform specific glyocosylation contributes to stabilization of ATP7A copper ATPase in plasma membranes.
The gastric H(+),K(+)-ATPase is an ATP-driven proton pump responsible for generating a million-fold proton gradient across the gastric membrane. We present the structure of gastric H(+),K(+)-ATPase at 6.5 A resolution as determined by electron crystallography of two-dimensional crystals. The structure shows the catalytic alpha-subunit and the non-catalytic beta-subunit in a pseudo-E(2)P conformation. Different from Na(+),K(+)-ATPase, the N-terminal tail of the beta-subunit is in direct contact with the phosphorylation domain of the alpha-subunit. This interaction may hold the phosphorylation domain in place, thus stabilizing the enzyme conformation and preventing the reverse reaction of the transport cycle. Indeed, truncation of the beta-subunit N-terminus allowed the reverse reaction to occur. These results suggest that the beta-subunit N-terminus prevents the reverse reaction from E(2)P to E(1)P, which is likely to be relevant for the generation of a large H(+) gradient in vivo situation.
The human multidrug resistance P-glycoprotein (P-gp) interacts with a broad range of compounds with diverse structures and
sizes. There is considerable evidence indicating that residues in transmembrane segments 4–6 and 10–12 form the drug-binding
site. We attempted to measure the size of the drug-binding site by using thiol-specific methanethiosulfonate (MTS) cross-linkers
containing spacer arms of 2 to 17 atoms. The majority of these cross-linkers were also substrates of P-gp, because they stimulated
ATPase activity (2.5- to 10.1-fold). 36 P-gp mutants with pairs of cysteine residues introduced into transmembrane segments
4–6 and 10–12 were analyzed after reaction with 0.2 mm MTS cross-linker at 4 °C. The cross-linked product migrated with lower mobility than native P-gp in SDS gels. 13 P-gp mutants
were cross-linked by MTS cross-linkers with spacer arms of 9–25 Å. Vinblastine and cyclosporin A inhibited cross-linking.
The emerging picture from these results and other studies is that the drug-binding domain is large enough to accommodate compounds
of different sizes and that the drug-binding domain is “funnel” shaped, narrow at the cytoplasmic side, at least 9–25 Å in
the middle, and wider still at the extracellular surface.
The free energy governing K(+) conduction through gramicidin A channels is characterized by using over 0.1 micros of all-atom molecular dynamics simulations with explicit solvent and membrane. The results provide encouraging agreement with experiments and insights into the permeation mechanism. The free energy surface of K(+), as a function of both axial and radial coordinates, is calculated. Correcting for simulation artifacts due to periodicity and the lack of hydrocarbon polarizability, the calculated single-channel conductance for K(+) ions is 0.8 pS, closer to experiment than any previous calculation. In addition, the estimated single ion dissociation constants are within the range of experimental determinations. The relatively small free energy barrier to ion translocation arises from a balance of large opposing contributions from protein, single-file water, bulk electrolyte, and membrane. Mean force decomposition reveals a remarkable ability of the single-file water molecules to stabilize K(+) by -40 kcal/mol, roughly half the bulk solvation free energy. The importance of the single-file water confirms the conjecture of Mackay et al. [Mackay, D. H. J., Berens, P. H., Wilson, K. R. & Hagler, A. T. (1984) Biophys. J. 46, 229-248]. Ion association with the channel involves gradual dehydration from approximately six to seven water molecules in the first shell, to just two inside the narrow pore. Ion permeation is influenced by the orientation of the single-file water column, which can present a barrier to conduction and give rise to long-range coupling of ions on either side of the pore. Small changes in the potential function, including contributions from electronic polarization, are likely to be sufficient to obtain quantitative agreement with experiments.
The sarcoplasmic reticulum Ca2+-ATPase, a P-type ATPase, has a critical role in muscle function and metabolism. Here we present functional studies and three new crystal structures of the rabbit skeletal muscle Ca2+-ATPase, representing the phosphoenzyme intermediates associated with Ca2+ binding, Ca2+ translocation and dephosphorylation, that are based on complexes with a functional ATP analogue, beryllium fluoride and aluminium fluoride, respectively. The structures complete the cycle of nucleotide binding and cation transport of Ca2+-ATPase. Phosphorylation of the enzyme triggers the onset of a conformational change that leads to the opening of a luminal exit pathway defined by the transmembrane segments M1 through M6, which represent the canonical membrane domain of P-type pumps. Ca2+ release is promoted by translocation of the M4 helix, exposing Glu 309, Glu 771 and Asn 796 to the lumen. The mechanism explains how P-type ATPases are able to form the steep electrochemical gradients required for key functions in eukaryotic cells.
The aqueous ionic radii of 35 ions have been calculated from published data of the average distances between the ions and the nearest water molecules, obtained by diffraction and computer simulation methods.
We show how a probabilistic interpretation of non-equilibrium thermodynamics, based on the assumption of local equilibrium at the mesoscale, can be used to analyze the stochastic dynamics of entropy driven diffusion processes characterized by the presence of entropic barriers. This approach sets up a systematic method to study the effect of confinement on the transport properties, providing a derivation of a generalized Fick–Jacobs equation for the constrained dynamics of the mesoscopic degrees of freedom. It is shown that confinement originates an entropic bias which gives rise to a geometric rectification of non-equilibrium fluctuations and that entropic effects in transport can be controlled by means of the application of an external force.
P-type ATPases form a large superfamily of cation and lipid pumps. They are remarkably simple with only a single catalytic subunit and carry out large domain motions during transport. The atomic structure of P-type ATPases in different conformations, together with ample mutagenesis evidence, has provided detailed insights into the pumping mechanism by these biological nanomachines. Phylogenetically, P-type ATPases are divided into five subfamilies, P1-P5. These subfamilies differ with respect to transported ligands and the way they are regulated.
Transport of spherical Brownian particles of finite size possessing radii R ≤ R(max) through narrow channels with varying cross-section width is considered. Applying the so-called Fick-Jacobs approximation, i.e. assuming fast equilibration in the direction orthogonal to the channel axis, the 2D problem can be described in terms of a 1D effective dynamics in which bottlenecks cause entropic barriers. Geometrical confinements result in entropic barriers which the particles have to overcome in order to proceed in the transport direction. The analytic findings for the nonlinear mobility for the transport are compared with precise numerical simulation results. The dependence of the nonlinear mobility on the particle size exhibits a striking resonance-like behavior as a function of the relative particle size ρ = R/R(max); this latter feature renders possible new effective particle separation scenarios.
The Na(+),K(+)-ATPase pump achieves thermodynamically uphill exchange of cytoplasmic Na(+) ions for extracellular K(+) ions by using ATP-mediated phosphorylation, followed by autodephosphorylation, to power conformational changes that allow ion access to the pump's binding sites from only one side of the membrane at a time. Formally, the pump behaves like an ion channel with two tightly coupled gates that are constrained to open and close alternately. The marine agent palytoxin disrupts this coupling, allowing both gates to sometimes be open, so temporarily transforming a pump into an ion channel. We made a cysteine scan of Na(+),K(+)-ATPase transmembrane (TM) segments TM1 to TM6, and used recordings of Na(+) current flow through palytoxin-bound pump-channels to monitor accessibility of introduced cysteine residues via their reaction with hydrophilic methanethiosulfonate (MTS) reagents. To visualize the open-channel pathway, the reactive positions were mapped onto a homology model of Na(+),K(+)-ATPase based on the structure of the related sarcoplasmicand endoplasmic-reticulum (SERCA) Ca(2+)-ATPase in a BeF(3)(-)-trapped state,(1,2) in which the extra-cytoplasmic gate is wide open (although the cytoplasmic access pathway is firmly shut). The results revealed a single unbroken chain of reactive positions that traverses the pump from the extracellular surface to the cytoplasm, comprises residues from TM1, TM2, TM4 and TM6, and passes through the equivalent of cation binding site II in SERCA, but not through site I. Cavity search analysis of the homology model validated its use for mapping the data by yielding a calculated extra-cytoplasmic pathway surrounded by MTS-reactive residues. As predicted by previous experimental results, that calculated extra-cytoplasmic pathway abruptly broadens above residue T806, at the outermost end of TM6 that forms the floor of the extracellular-facing vestibule. These findings provide a structural basis for further understanding cation translocation by the Na(+),K(+)-ATPase and by other P-type pumps like the Ca(2+)- and H(+),K(+)-ATPases.
Coefficients for active transport of ions and heat in vesicles with Ca(2+)-ATPase from sarcoplasmic reticulum are defined in terms of a newly proposed thermodynamic theory and calculated using experiments reported in the literature. The coefficients characterize in a quantitative manner different performances of the enzyme isoforms. Four enzyme isoforms are examined, namely from white and red muscle tissue, from blood platelets, and from brown adipose mitochondria. The results indicate that the isoforms have a somewhat specialized function. White muscle tissue and brown adipose tissue have the same active transport coefficient ratio, but the activity level of the enzyme in white muscle is higher than in brown adipose tissue. The thermogenesis ratio is high in both white muscle and brown adipose tissue, in agreement with a specific role in nonshivering thermogenesis. Other isoforms do not have this ability to generate heat. A calcium-dependence of the coefficients is found, which can be understood as being in accordance with the role of this ion as a messenger in muscle contraction as well as in thermogenesis. The investigation points to new experiments related to structure as well as to function of the isoforms.
We present a new mesoscopic basis which can be used to derive flux equations for the forward and reverse mode of operation of ion-pumps. We obtain a description of the fluxes far from global equilibrium. An asymmetric set of transport coefficients is obtained, by assuming that the chemical reaction as well as the ion transports are activated, and that the enzyme has a temperature independent of the activation coordinates. Close to global equilibrium, the description reduces to the well known one from non-equilibrium thermodynamics with a symmetric set of transport coefficients. We show how the measurable heat flux and the heat production under isothermal conditions, as well as thermogenesis, can be defined. Thermogenesis is defined via the onset of the chemical reaction or ion transports by a temperature drop. A prescription has been given for how to determine transport coefficients on the mesocopic level, using the macroscopic coefficient obtained from measurements, the activation enthalpy, and a proper probability distribution. The method may give new impetus to a long-standing unsolved transport problem in biophysics.
We use the mesoscopic nonequilibrium thermodynamics theory to derive the general kinetic equation of a system in the presence of potential barriers. The result is applied to a description of the evolution of systems whose dynamics is influenced by entropic barriers. We analyze in detail the case of diffusion in a domain of irregular geometry in which the presence of the boundaries induces an entropy barrier when approaching the exact dynamics by a coarsening of the description. The corresponding kinetic equation, named the Fick-Jacobs equation, is obtained, and its validity is generalized through the formulation of a scaling law for the diffusion coefficient which depends on the shape of the boundaries. The method we propose can be useful to analyze the dynamics of systems at the nanoscale where the presence of entropy barriers is a common feature.
The time-resolved kinetics of Ca2+ binding to the SR Ca-ATPase in the E1 state was investigated by Ca(2+)-concentration jump experiments. Ca2+ was released by an ultraviolet-light flash from caged calcium, and charge movements in the membrane domain of the ion pumps were detected by the fluorescent styryl dye 2BITC. The partial reaction (H3E1 <-->) E1 <--> CaE1 <--> Ca2E1 can be characterized by two time constants, tau1 and tau2, both of which are not significantly Ca(2+)-concentration-dependent and only weakly pH-dependent at pH < 7.5. Both time constants differ by a factor of approximately 50 (4.7 vs. 200 ms). The weak substrate-dependence indicates that the rate-limiting process is not related to Ca2+ migration through the access channel and ion binding to the binding sites but to conformational rearrangements preceding the ion movements. The high activation energy obtained for both processes, 42.3 kJ mol(-1) and 60.3 kJ mol(-1) at pH 7.2, support this concept. Transient binding of Ca ions to the loop L67 and a movement of the Ca-loaded loop are discussed as a mechanism that facilitates the entrance of both Ca ions into the access channel to the ion-binding sites.
We study biased, diffusive transport of Brownian particles through narrow, spatially periodic structures in which the motion is constrained in lateral directions. The problem is analyzed under the perspective of the Fick-Jacobs equation, which accounts for the effect of the lateral confinement by introducing an entropic barrier in a one-dimensional diffusion. The validity of this approximation, based on the assumption of an instantaneous equilibration of the particle distribution in the cross section of the structure, is analyzed by comparing the different time scales that characterize the problem. A validity criterion is established in terms of the shape of the structure and of the applied force. It is analytically corroborated and verified by numerical simulations that the critical value of the force up to which this description holds true scales as the square of the periodicity of the structure. The criterion can be visualized by means of a diagram representing the regions where the Fick-Jacobs description becomes inaccurate in terms of the scaled force versus the periodicity of the structure.